Suggest to a Gary Johnson supporter, or a Jill Stein supporter, that voting for their preferred candidate is a wasted vote and you’re likely to get an ear full. But whether or not such a vote is actually a wasted vote depends on what you think of the other candidates.

As this 2016 presidential election has dragged on (and on) between two historically disliked front-running candidates, like campaign fingernails across a 16-month long chalkboard, heartfelt debates over how to vote are ubiquitous. And, as often as not, such debates get around to the idea of the wasted vote.

Everyone seems to know, intuitively, what constitutes a wasted vote. However, there is more nuance around this issue than many people consider. I’m reminded of a pithy quote, dating back to 1917:

“Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong.” — H. L. Menken

Consider the recent blog post, The Power of the Wasted Vote, that begins with a pithy quote:

“I’d rather vote for what I want and not get it, than vote for what I don’t want and get it.” — Eugene V. Debs

Upon initial glance, the Debs quote seems quite reasonable, even wise. But, in fact, this is a perfect example of the type of explanation Menken was talking about: neat, plausible, and wrong. Let’s look a little more carefully to see why this is so.

When there are only two alternatives (i.e., candidates), this statement is, in fact, reasonable. Consider two alternatives A (the one you want) and B (the one you don’t want). In this scenario, not getting A implies getting B. Now, the Debs quote becomes:

I’d rather vote for A and get B than vote for B and get B.

In this scenario, you’re getting alternative B, period. So the statement can be refactored as:

I’d rather vote for A than vote for B. (But I’m getting B, regardless).

This really just expresses your preference for A, which is a given. So, in this scenario, it makes sense to register your preference for alternative A, in what might be called a protest vote, not because it will affect the outcome of this election, but because it might influence what happens after the election. (Don’t get hung up on the “protest vote” label. It’s just a label. If you feel that has a negative connotation, use any other label you prefer.)

The two-alternative scenario is easily analyzed. But the logic quickly becomes more complicated–less “neat”–when there are more than two alternatives.

To model the case of additional alternatives, we’ll continue to treat alternative A as most preferred. But now we’ll treat B as a list of alternatives B1, B2, …, all of which you don’t want. Now, as before, you’ll be ending up with B, but in this case, that means you’ll be getting one of the B1, B2, …. alternatives. In this case, a “protest vote” makes sense only if you have no significant preference between the B1, B2, … alternatives.

To see this, let’s make the example more concrete:

A = Getting kissed on the cheek by your biggest crush

B1 = Getting kissed on the cheek by your sibling

B2 = Getting kissed on the cheek by your other sibling

Now, following the logic of the Debs quote, you would vote for getting kissed by your biggest crush. However, you’ll actually end up getting kissed by one of your siblings, and you’ll leave that choice up to the other voters. This is okay because you don’t have a significant preference between B1 and B2.

A fundamental problem with the protest vote logic is when it ignores relative preferences between the available candidates. Let’s tweak the alternatives as follows

A = Getting kissed on the cheek by your biggest crush

B1 = Getting smacked in the face with a plastic ruler

B2 = Getting smacked in the face with a sledgehammer

We still most prefer A, but now, there is a strong preference between B1 and B2. In this case, it is very risky to leave the B1 vs. B2 choice entirely up to the other voters. In this case, if the race between B1 and B2 is at all close, a protest vote (i.e., a vote for A) is a wasted vote! It is wasted because it could have been used to influence a choice (B1 vs. B2) in which you have a strong preference. Put another way, the protest vote did not have maximum utility.

We can, and generally do, have more than one preference. In the example above we have the following preferences:

A > B1

A > B2

B1 > B2

And there is a fourth, less obvious preference:

We’d like to publicly register both A > B1 and A > B2, in the hopes to influence others, now or in the future.

But we only get to cast our vote once, for a single alternative. We should cast it in a way that maximizes its utility. That gets the most bang for the buck, so to speak. In order to do this, we must consider the relative strengths of each preference.

So, here’s my attempt to boil this whole idea down to my own pithy quote:

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6 comments for “When is a Vote a Wasted Vote?”

Don Kinzer

October 28, 2016 at 10:38 pm

Your analysis is reasonable as far as it goes. However, the factor that it omits is the likelihood of a candidate actually being elected. In this year’s contest, either Trump or Clinton will prevail (barring an extreme unforeseen event) so a vote for anyone else is, effectively, wasted. If the probabilities of being elected between Johnson, Stein, Clinton and Trump were relatively close then it would be difficult to ascertain, lacking other information, if a vote for any one candidate would be wasted.

The decision of voters who are thinking rationally probably comes down to which of the two (Trump or Clinton) is the least offensive to their particular sensibilities. If one disfavors Trump or Clinton then voting for anyone other than Clinton or Trump, respectively, is thus a wasted vote.

You’re right, Don. I did not clearly address the issue of probabilities, and in general, one should consider the probability of each outcome. (I actually started down that path, but I realized I’d double the size of a post that was already longer than most people like to read. )

By focusing just on the Debs quote, probabilities are partially considered–indirectly–because we already know that alternative A will not win. So, p(A)==0. This means that a vote for A is wasted to the extent that you actually have a preference between the (likely) alternatives.

I suspect we could formulate a “wastefulness” probability function that would be the likelihood that a vote for alternative x was wasted, as a function of two vectors: 1) the probabilities of each alternative winning, and b) your personal preference distribution across all alternatives.

The way I see it, the only wasted vote is one that is ignorant or dishonest. Voting isn’t a bet. The point isn’t to pick a winner. Voting asks each of us to use our best judgment in answering the question, “Who would make the best President?” If we fail to answer that question honestly, if we instead game the system in order to manipulate the outcome, we are actively undermining democracy.

That’s a reasonable interpretation. But, I don’t think that’s exactly the question. As I see it, a more accurate question would be “Given everything you know about this slate of candidates, including what you know about how other voters are likely to vote, to which candidate do you want to give your one vote?” In that phrasing, you’re not (necessarily) voting for you most preferred candidate. Rather, you’re voting to influence the outcome of the election. That is a subtly different thing.

Utter crap. Nowhere do you talk about voting blue in Idaho or red in California. Instead you pick on third parties. When one votes, they are endorsing someone, not trying to influence an election. The whole idea of a wasted vote is a shame invented by the two sides coin big parties of your country. Stick to worrying about your own vote & leave the rest alone.

Elite Liberal, thank you for your comment. Here are my thoughts: It may or may not be the case that when someone votes, they are intending to influence the election. When I vote, I am. Perhaps, when you vote, you are not. But, regardless of one’s intent, when we vote, we are influencing the election.

Now, if a voter truly does not care about the outcome of the election, and cares only about registering an endorsement of a particular candidate, then I completely agree with you. In that case, then one’s vote cannot be wasted. If on the other hand, a voter does care about the outcome of the election and performs a game-theoretical analysis of the most effective use of his or her one vote, then it is quite possible for a vote to be wasted.

When trying to influence an election–as I and many other voters do–it would be irrational to not consider the probable votes cast by others when deciding how to cast your own.